A Mathematical Design of Genetic Operators on 𝐺𝐿_𝑛(ℤ₂)

Algorithm 1

An algorithm for repairing offspring generated by typical 2D approach.

RepairBinaryMatrix(𝑀=(𝑚_𝑖𝑗))

{

 // Let 𝑚_𝑖 be the 𝑖th row vector of 𝑀, that is, 𝑀^𝑇=𝑚₁𝑚₂…𝑚_𝑛.

 // Let 𝐼 be the identity matrix, where 𝐼^𝑇=𝑒₁𝑒₂…𝑒_𝑛.

 𝐽←{1,2,…,𝑛};

 𝐾←∅;

 𝑅←∅;

 for  each 𝑚_𝑖, 𝑖∈𝐽 // use the process of Gauss-Jordan elimination

  if  𝑚_𝑖≠0 and 𝑚_𝑖 is not spanned by 𝑅  then

   𝑅←𝑅∪{𝑚_𝑖};

   𝐾←𝐾∪{𝑖};

 // Then, 𝑅 becomes the set of maximal linearly-independent row vectors.

 𝑆←∅;

 for  each 𝑒_𝑖, 𝑖∈𝐽

  if  𝑒_𝑖 is not spanned by 𝑅  then

   𝑆←𝑆∪{𝑒_𝑖};

 for each 𝑖∈𝐽−𝐾

  𝑟← any element in 𝑆;

  𝑆←𝑆−{𝑟};

  𝑚_𝑖←𝑟; // repairing 𝑚_𝑖

 return  𝑀=(𝑚_𝑖𝑗);

}